I want to whether there is a vector generating function (/matrix) such that it can generate a m-dimensional vector which will always be linearly independent of the set of m-dimensional vectors the function has already generated.

My problem can be written in pseudocode format as follow. I therefore expect that any m randomly picked vectors from the pool of the N vectors will generate a full-rank matrix.

Vandermonde matrix is one possible option, but it requires the use of exponentially large field size. So I am looking for vectors generated over smaller field size. Any help in this direction will be greatly appreciated.

1 Answer
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I don't quite follow the first paragraph of your question. But reading the rest of your post, if you only need a set $S$ of $N$ $m$-dimensional vectors over the finite field of some small order in which any subset $S' \subset S$ of cardinality $m$ is a set of linearly independent vectors, that's a parity-check matrix of an MDS code over $\mathbb{F}_q$.

The binary case is no good because the only MDS codes are the trivial ones. For larger $q$, there are known nontrivial MDS codes. Reed-Solomon codes are good examples. They're cyclic codes so you can realize the codewords as an ideal of the polynomial ring $\mathbb{F}_q[x]/(x^N-1)$; you can generate them through multiplication between a certain monic polynomial that divides $x^N-1$ and all polynomials of degree less than $N-m$ (including $0$). Maybe this is systematic enough to work for your purpose?

In any case, you might want to clarify your question a bit or, if MDS codes are indeed what you would like to construct, pick your favorite coding theory textbook and read up on them.